Independent Events Based on a Pitney Bowes survey, when 1009 consumers were asked if they are comfortable with drones delivering their purchases, 42% said yes. Consider the probability that among 30 different consumers randomly selected from the 1009 who were surveyed, there are at least 10 who are comfortable with the drones. Given that the subjects surveyed were selected without replacement, are the 30 selections independent? Can they be treated as being independent? Can the probability be found by using the binomial probability formula? Explain.

It is given that 42% of consumers are comfortable having drones deliver their purchases and the rest are not.

From a sample of 1009 consumers who were surveyed, a group of 30 consumers isselected, and out of those 30, the probability of getting at least 10 who are comfortable is to be found.

The selections are made without replacement.

When selections are made without replacement, they are treated as dependent events.

Therefore, the 30 selections made are not independent.

By using the 5% rule, it can be examined whether the selections can be treated as independent or not.

The rule states that if the sample size chosen is no more than 5% of the population size, then the individual selections can be assumed to be independent even if the selections are made without replacement.

Here, the population size is equal to 1009.

The sample size is equal to 30.

The following computations are made to examine if the rule holds good:

$$\begin{gathered} 5\%\text{of}1009=\frac{5}{100}\times 1009 \\ =50.49 \end{gathered}$$

It can be seen that $30<50.49$.

Since the sample size is no more than 5% of the population size, the selections can be assumed to be independent.

The key requirement for a variable following the binomial probability distribution is that the individual selections/events should be independent.

Since the above assumption is fulfilled, it can be said that the probability of selecting at least 10 out of 30 consumers who are comfortable with drones can be computed using the binomial probability formula.

Here, let X denote the number of consumers who are comfortable with drones.

The probability of getting at least 10 consumers who are comfortable with drones has the following expression:

$P\left(X⩾10\right)=1-P\left(X<10\right)$

Here, the formula needs to be iterated 10 times which can make the calculations complex and lengthy.

Thus, the required probability can be computed easily using technology as compared to applying the formula manually.